Elliptic triangle groups in , Lagrangian triples and momentum maps

We determine the possible eigenvalues of elliptic matrices A , B , C in P U ( 2 , 1 ) satisfying A B C = 1 . This is done by describing geometrically the image of a group-valued momentum map for the (non-compact) group action of P U ( 2 , 1 ) by conjugation on C 1 × C 2 where C 1 and C 2 are fixed elliptic conjugacy classes in P U ( 2 , 1 ) . Contrary to the compact case, this image is not always convex; rather it is the union of one, two or three convex polygons in T 2 / S 2 . The main motivation was to analyze elliptic triangle groups in P U ( 2 , 1 ) such as Mostow’s lattices.